Abstract

We discuss a variety of iterative methods that are based on the
Arnoldi process for solving large sparse symmetric indefinite linear
systems. We describe the SYMMLQ and SYMMQR methods, as
well as generalizations and modifications of them. Then, we cover
the Lanczos/MSYMMLQ and Lanczos/MSYMMQR methods, which
arise from a double linear system. We present pseudocodes for these
algorithms.

The authors dedicate this paper to the memory of Professor David M. Young, Jr., for his pioneering research, inspirational teaching, and exceptional life

1. Introduction

Frequently, when computing numerical solutions of partial differential equations, one needs to solve systems of very large sparse linear algebraic equations of the form
where is an matrix, is an vector, and one seeks a numerical solution vector or a good approximation of it. Particularly for large linear systems arising from partial differential equations in three dimensions, well-known direct methods, such as Gaussian elimination, may become prohibitively expensive in terms of both computer storage and computer time. On the other hand, a variety of iterative methods may avoid these difficulties.

For linear systems involving symmetric positive definite (SPD) matrices, the conjugate gradient (CG) method (and variations of it) may work well. On the other hand, when solving linear systems, where the coefficient matrix is symmetric indefinite, the choice of a suitable iterative method is not at all clear. On the other hand, the SYMMLQ and MINRES methods have been shown to be useful in certain situations (see Paige and Saunders [1]). For nonsymmetric systems, Saad and Schultz [2] generalized the MINRES method to obtain the GMRES method.

In Section 2, we review the Arnoldi process. In Sections 3 and 4, we describe the SYMMLQ and SYMMQR methods. Then we can generalize them, in Section 5, and we outline the modified SYMMLQ method, in Section 6. Next, in Section 7, we discuss applying the MSYMMLQ and MSYMMQR methods applied to a double linear system. Finally, we present pseudocodes in Sections 8–11.

2. Arnoldi Process

We begin with a review of the Arnoldi process.

Theorem 1. Suppose that is an symmetric matrix. One can generate orthonormal vectors using this short-term recurrence
where
Here, one assumes that and , for all . Then the following properties hold, for (, ):

Proof. If we let , then the subspace
is equivalent to the Krylov subspace
We obtain
since .

From (2) and (3), we have
Consequently, we obtain, since ,
So we obtain

3. SYMMLQ Method

We choose , such that . Hence, we have

Imposing the Galerkin condition , we obtain
We obtain
because
Instead of solving for directly from the triangular linear system (15), Paige and Saunders [1] factorize the matrix into a lower triangular matrix with bandwidth three (resulting in the SYMMLQ method). Also, we have
where is an orthogonal matrix, and
where . Since , we have
Letting
then
Next letting
we have
Defining
we have
where
We let
where
From (21) and (28), we have . Since
we have
If , then is nonsingular. We can find by solving

4. SYMMQR Method

We choose such that . Hence, we have
Imposing the Galerkin condition , as before, we obtain
Since
we have
Instead of solving for directly from the triangular system (35), Paige and Saunders [1] factorized the matrix into a lower triangular matrix with bandwidth three.

We can use a different factorization of to obtain a slightly different method, which is called the SYMMQR method. We multiply the matrix by an orthogonal matrix on the left-hand side instead of the right-hand side. We have
where
We obtain the matrix , where
with being the Givens rotation. Letting be the solution of
then we have
which satisfies the Galerkin condition , where . We note that is not always nonzero and, thus, might be singular. We assume that is nonsingular and then we define
where
We have

For the next iterate , we need to solve
where
Applying the Givens rotation to both sides of (45), we have
where .

To eliminate , we compute the th Given rotation by
By multiplying times and times , we have
where
Let
We define . Since , then and is nonsingular. We can solve for from
We discuss the case later.

Consider solving the least square problem involving minimizing , where
We have
Hence, the solution from minimizes and .

Let
where
We have
Since
we obtain
We note that is the estimated solution vector satisfying the Galerkin condition, while
with minimizing .

5. Generalized SYMMLQ and SYMMQR Methods

Now, we generalize the SYMMLQ and SYMMQR methods.

Theorem 3. Suppose that is an symmetric positive definite (SPD) matrix and is an symmetric matrix. One can generate orthonormal vectors using this short-term recurrence
where
Then the following properties hold, for (, ):

Proof. We obtain
Since .

As before, we let
Moreover, we have
where

As before, we let
Imposing the Galerkin condition again, we have
We obtain
because
Since is symmetric, we can apply the same techniques as in the SYMMLQ method. Also, if , the method reduces to the SYMMQR method.

6. Modified SYMMLQ Method

Next, we outline the modified SYMMLQ method.

Theorem 4. Suppose that is an symmetric (not necessary positive definite) matrix and is an symmetric matrix. One can generate orthonormal vectors using this short-term recurrence
where
Then the following properties hold, for :
and, for (, ),

From Theorem 4, in matrix form, we obtain
where
Moreover, from Theorem 4, we obtain
Then, we have
Here the second term on the right-hand side is the zero matrix!

In addition, we have
Imposing the Galerkin condition, , as we did before, we obtain
In other words, we use
We obtain
because
HereWe note that is symmetric, for :

7. Lanczos/MSYMMLQ Method

Next, we consider this double linear system:
We obtain the block symmetric matrices , , and , where

For example, the modified SYMMLQ method and the modified SYMMQR method can be applied to the double linear system (86). This leads us to the LAN/MSYMMLQ method and the LAN/MSYMMQR method. The pseudocodes for these methods are given in the following sections. For additional details, see Li [3]. See the books by Golub and Van Loan [4] and Saad [5], as well as the papers by Lanczos [6] and Kincaid et al. [7], among others.

8. MSYMMLQ Pseudocode

9. MSYMMQR Pseudocode

10. LAN/MSYMMLQ Pseudocode

11. LAN/MSYMMQR Pseudocode

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

C. Lanczos, “An iteration method for the solution of the eigenvalue problem of linear differential and integral operators,” Journal of Research of the National Bureau of Standards, vol. 45, no. 4, pp. 255–282, 1950.View at Google Scholar